トップ› 北海道大学› 2010年› 理系第5問

北海道大学 2010年理系第5問解説

譁ｹ驥昴・蛻晄焔

縺上§繧貞ｼ輔″邯壹￠繧倶ｺ玖ｱ｡繧偵√・02蝗槫・縺ｮ縺上§縺ｮ荳ｦ縺ｳ鬆・阪ｒ豎ｺ螳壹☆繧句撫鬘後→縺励※謐峨∴縺ｾ縺吶ょ・莠玖ｱ｡繧偵・02蛟九・蝣ｴ謇縺九ｉ縲・縺､縺ｮ蠖薙◆繧翫￥縺倥′蜈･繧句ｴ謇繧帝∈縺ｶ邨・粋縺帙阪→閠・∴縲√◎繧後ｉ縺悟酔讒倥↓遒ｺ縺九ｉ縺励＞縺ｨ縺励※遒ｺ邇・ｒ險育ｮ励＠縺ｦ縺・￥縺ｮ縺悟渕譛ｬ譁ｹ驥昴〒縺吶・

隗｣豕・

(1)

102譛ｬ縺ｮ縺上§繧貞ｼ輔￥鬆・分繧偵☆縺ｹ縺ｦ豎ｺ螳壹☆繧九→縺阪・譛ｬ縺ｮ蠖薙◆繧翫￥縺倥・菴咲ｽｮ繧帝∈縺ｶ蝣ｴ蜷医・謨ｰ縺ｯ縲・ $$ {}_{102}\mathrm{C}_{2} = \frac{102 \cdot 101}{2} = 5151 $$ 騾壹ｊ縺ｧ縺ゅｊ縲√％繧後ｉ縺ｯ蜷梧ｧ倥↓遒ｺ縺九ｉ縺励＞縲・

$n$ 蝗樒岼縺ｫ1譛ｬ逶ｮ縺ｮ蠖薙◆繧翫￥縺倥′蜃ｺ繧九◆繧√・譚｡莉ｶ縺ｯ縲・n$ 蝗樒岼縺悟ｽ薙◆繧翫￥縺倥〒縺ゅｊ縲√°縺､谿九ｊ縺ｮ1譛ｬ縺ｮ蠖薙◆繧翫￥縺倥′ $n+1$ 蝗樒岼縺九ｉ $102$ 蝗樒岼縺ｮ縺・★繧後°縺ｫ蜃ｺ繧九％縺ｨ縺ｧ縺ゅｋ縲・縺励◆縺後▲縺ｦ縲・n$ 縺ｯ $1 \leqq n \leqq 101$ 縺ｧ縺ゅｊ縲∵ｮ九ｊ縺ｮ蠖薙◆繧翫￥縺・譛ｬ縺ｮ菴咲ｽｮ縺ｮ驕ｸ縺ｳ譁ｹ縺ｯ $102 - n$ 騾壹ｊ蟄伜惠縺吶ｋ縲・

繧医▲縺ｦ縲∵ｱゅａ繧狗｢ｺ邇・・縲・ $$ \frac{102 - n}{5151} $$ 縺ｧ縺ゅｋ縲・

(2)

(1)縺ｧ豎ゅａ縺・$n$ 蝗樒岼縺ｫ1譛ｬ逶ｮ縺ｮ蠖薙◆繧翫￥縺倥′蜃ｺ繧狗｢ｺ邇・ｒ $p_n$ 縺ｨ縺吶ｋ縲・ A, B, C 縺ｮ3莠ｺ縺ｯ鬆・分縺ｫ縺上§繧貞ｼ輔￥縺溘ａ縲√◎繧後◇繧後′縺上§繧貞ｼ輔￥縲悟屓逶ｮ縲阪→蝗樊焚縺ｯ莉･荳九・繧医≧縺ｫ縺ｪ繧九ょ・菴薙〒102蝗槫ｼ輔￥縺溘ａ縲・莠ｺ縺ｯ縺昴ｌ縺槭ｌ縺｡繧・≧縺ｩ34蝗槭★縺､縺上§繧貞ｼ輔￥縲・

A縺悟ｼ輔￥縺ｮ縺ｯ縲・n = 3k - 2 \quad (k = 1, 2, \dots, 34)$ 縺ｮ縺ｨ縺阪・
B縺悟ｼ輔￥縺ｮ縺ｯ縲・n = 3k - 1 \quad (k = 1, 2, \dots, 34)$ 縺ｮ縺ｨ縺阪・
C縺悟ｼ輔￥縺ｮ縺ｯ縲・n = 3k \quad (k = 1, 2, \dots, 34)$ 縺ｮ縺ｨ縺阪・

1譛ｬ逶ｮ縺ｮ蠖薙◆繧翫￥縺倥ｒA縺悟ｼ輔￥遒ｺ邇・・縲・p_n$ 繧・$k=1$ 縺九ｉ $34$ 縺ｾ縺ｧ雜ｳ縺怜粋繧上○繧九％縺ｨ縺ｧ豎ゅａ繧峨ｌ繧九・ $$ \begin{aligned} \sum_{k=1}^{34} p_{3k-2} &= \sum_{k=1}^{34} \frac{102 - (3k - 2)}{5151} \\ &= \frac{1}{5151} \sum_{k=1}^{34} (104 - 3k) \\ &= \frac{1}{5151} \left( 104 \cdot 34 - 3 \cdot \frac{34 \cdot 35}{2} \right) \\ &= \frac{1}{5151} (3536 - 1785) \\ &= \frac{1751}{5151} \end{aligned} $$ 縺薙％縺ｧ縲・1751 = 17 \cdot 103$縲・5151 = 51 \cdot 101 = 17 \cdot 303$ 縺ｧ縺ゅｋ縺九ｉ縲・ $$ \frac{1751}{5151} = \frac{103}{303} $$

蜷梧ｧ倥↓縲。縺悟ｼ輔￥遒ｺ邇・・縲・ $$ \begin{aligned} \sum_{k=1}^{34} p_{3k-1} &= \sum_{k=1}^{34} \frac{102 - (3k - 1)}{5151} \\ &= \frac{1}{5151} \sum_{k=1}^{34} (103 - 3k) \\ &= \frac{1}{5151} \left( 103 \cdot 34 - 3 \cdot \frac{34 \cdot 35}{2} \right) \\ &= \frac{1}{5151} (3502 - 1785) \\ &= \frac{1717}{5151} \end{aligned} $$ 縺薙％縺ｧ縲・1717 = 17 \cdot 101$ 縺ｧ縺ゅｋ縺九ｉ縲・ $$ \frac{1717}{5151} = \frac{17 \cdot 101}{17 \cdot 303} = \frac{101}{303} = \frac{1}{3} $$

蜷梧ｧ倥↓縲，縺悟ｼ輔￥遒ｺ邇・・縲・ $$ \begin{aligned} \sum_{k=1}^{34} p_{3k} &= \sum_{k=1}^{34} \frac{102 - 3k}{5151} \\ &= \frac{1}{5151} \sum_{k=1}^{34} (102 - 3k) \\ &= \frac{1}{5151} \left( 102 \cdot 34 - 3 \cdot \frac{34 \cdot 35}{2} \right) \\ &= \frac{1}{5151} (3468 - 1785) \\ &= \frac{1683}{5151} \end{aligned} $$ 縺薙％縺ｧ縲・1683 = 17 \cdot 99$ 縺ｧ縺ゅｋ縺九ｉ縲・ $$ \frac{1683}{5151} = \frac{17 \cdot 99}{17 \cdot 303} = \frac{99}{303} = \frac{33}{101} $$

隗｣豕・

**(2)縺ｮ蛻･隗｣・域焚蛻励・諤ｧ雉ｪ繧貞茜逕ｨ縺吶ｋ隗｣豕包ｼ・*

(1)繧医ｊ縲・n$ 蝗樒岼縺ｫ1譛ｬ逶ｮ縺ｮ蠖薙◆繧翫ｒ蠑輔￥遒ｺ邇・$p_n$ 縺ｯ縲・ $$ p_n = \frac{102 - n}{5151} $$ 縺ｧ縺ゅｊ縲√％繧後・ $n$ 縺ｮ1谺｡蠑上〒縺ゅｋ縺溘ａ縲∵焚蛻・$\{ p_n \}$ 縺ｯ遲牙ｷｮ謨ｰ蛻励ｒ縺ｪ縺吶・

A, B, C 縺・譛ｬ逶ｮ縺ｮ蠖薙◆繧翫ｒ蠑輔￥遒ｺ邇・ｒ縺昴ｌ縺槭ｌ $P(A), P(B), P(C)$ 縺ｨ縺吶ｋ縺ｨ縲√◎繧後◇繧後′蠑輔￥蝗樒岼縺ｯ隗｣豕・縺ｨ蜷梧ｧ倥〒縺ゅｊ縲・ $$ \begin{aligned} P(A) &= p_1 + p_4 + \dots + p_{100} \\ P(B) &= p_2 + p_5 + \dots + p_{101} \\ P(C) &= p_3 + p_6 + \dots + p_{102} \end{aligned} $$ 縺薙ｌ繧峨・縺昴ｌ縺槭ｌ鬆・焚 $34$ 縺ｮ蜥後〒縺ゅｋ縲・蜷・・・蟾ｮ縺ｫ豕ｨ逶ｮ縺吶ｋ縺ｨ縲・ $$ p_{3k-2} - p_{3k-1} = \frac{1}{5151} $$ $$ p_{3k-1} - p_{3k} = \frac{1}{5151} $$ 縺ｧ縺ゅｋ縺九ｉ縲・ $$ P(A) - P(B) = \sum_{k=1}^{34} (p_{3k-2} - p_{3k-1}) = 34 \cdot \frac{1}{5151} = \frac{34}{5151} $$ $$ P(B) - P(C) = \sum_{k=1}^{34} (p_{3k-1} - p_{3k}) = 34 \cdot \frac{1}{5151} = \frac{34}{5151} $$ 縺ｨ縺ｪ繧九ゅ％繧後・ $P(A), P(B), P(C)$ 繧ゅ％縺ｮ鬆・↓遲牙ｷｮ謨ｰ蛻励ｒ縺ｪ縺吶％縺ｨ繧呈э蜻ｳ縺吶ｋ縲・縺励◆縺後▲縺ｦ縲∫ｭ牙ｷｮ謨ｰ蛻励・諤ｧ雉ｪ繧医ｊ縲・ $$ P(A) + P(C) = 2P(B) $$ 縺梧・繧顔ｫ九▽縲・縺ｾ縺溘・譛ｬ逶ｮ縺ｮ蠖薙◆繧翫￥縺倥・蠢・★隱ｰ縺九′蠑輔￥縺溘ａ縲・ $$ P(A) + P(B) + P(C) = 1 $$ 縺ｧ縺ゅｋ縲ゅ％繧後ｉ繧帝｣遶九＆縺帙ｋ縺ｨ縲・ $$ 3P(B) = 1 \iff P(B) = \frac{1}{3} $$ 縺薙ｌ繧堤畑縺・※ $P(A)$ 縺ｨ $P(C)$ 繧呈ｱゅａ繧九・ $$ \frac{34}{5151} = \frac{2 \cdot 17}{303 \cdot 17} = \frac{2}{303} $$ 縺ｧ縺ゅｋ縺九ｉ縲・ $$ P(A) = P(B) + \frac{2}{303} = \frac{1}{3} + \frac{2}{303} = \frac{101}{303} + \frac{2}{303} = \frac{103}{303} $$ $$ P(C) = P(B) - \frac{2}{303} = \frac{1}{3} - \frac{2}{303} = \frac{101}{303} - \frac{2}{303} = \frac{99}{303} = \frac{33}{101} $$

隗｣隱ｬ

縺上§繧帝・分縺ｫ蠑輔￥蝠城｡後〒縺ｯ縲∵擅莉ｶ繧呈ｺ縺溘☆蝣ｴ蜷医ｒ荳縺､荳縺､閠・∴繧九ｈ繧翫ｂ縲√≠繧峨°縺倥ａ蜈ｨ縺ｦ縺ｮ縺上§繧貞ｼ輔″蛻・ｋ鬆・・繧・ｵ・粋縺帙ｒ諠ｳ螳壹☆繧九％縺ｨ縺ｧ縲∬ｦ矩壹＠繧医￥隗｣縺上％縺ｨ縺後〒縺阪∪縺吶・ (2)縺ｧ縺ｯ縲∫｢ｺ邇・ｒ蜥瑚ｨ伜捷・・\Sigma$・峨ｒ逕ｨ縺・※諢夂峩縺ｫ險育ｮ励＠縺ｦ繧よｭ｣隗｣縺ｫ縺溘←繧顔捩縺代∪縺吶′縲・1)縺ｧ蠕励ｉ繧後◆遒ｺ邇・・蠑上′ $n$ 縺ｫ髢｢縺吶ｋ1谺｡蠑上〒縺ゅｋ縺薙→繧貞茜逕ｨ縺吶ｋ縺ｨ縲∫ｭ牙ｷｮ謨ｰ蛻励・諤ｧ雉ｪ縺九ｉ險育ｮ鈴㍼繧貞括逧・↓貂帙ｉ縺吶％縺ｨ縺後〒縺阪∪縺吶ゅ％繧後・謨ｰ蛻励→遒ｺ邇・′陞榊粋縺励◆蝠城｡後↓縺翫＞縺ｦ髱槫ｸｸ縺ｫ譛牙柑縺ｪ螳夂浹縺ｧ縺吶・